SINGULARLY
IMPORTANT
IMPORTANT
Keep tally of these scalars if you want to compute rank(A) = rank(U Sigma V^T)
What are "nonzero singular values"?
If "U" don't want the error to blow up, use this option when solving Ax = b "a-la Gauss"
What is "row pivoting"?
This value represents the difference between 1 and the floating point value closest to it, at least on your hardware
What is "machine epsilon"?
Let's get into shape: reduce A to this form before racing off to find eigenvalues
What is "(upper) Hessenburg"?
|x^Ty| leq ||x||*||y||
What is the "Cauchy-Schwarz inequality"?
I added up the eigenvalues of A, and all I got was this value, which is itself defined as a different sum altogether!
What is the "trace"?
The singular values of A in bbb C^n are the square roots of the eigenvalues of this related matrix
What is "A-star-A"?
Don't F-A-I-L to count these up when predicting the runtime of your algorithm!
What are "FLOPS"?
It takes this many bits to represent a single complex number in double-precision floating point
What is "128"?
This one "bullies" the others into submission when you apply the Power Method
What is the "dominant eigenvalue"?
(||x||*||J(x)||)/||f(x)||
This quantity is defined as sqrt(tr(A^**A)), although it would be unwise to actually compute it that way!
What is the "Frobenius norm"?
This SVD-derived ratio is another way of computing the matrix condition number
What is the "the biggest singular value divided by the smallest"?
Give the matrix a little "spin" by using this method to ortho-normalize its column vectors
What is "Givens rotations"?
The "laws" of floating point are laid down in a standard which bears this alphanumeric name
What is "IEEE 754"?
If A = QR, then this term refers to the relation which guarantees that A and RQ have the same eigenvalues
What is "(unitary) similarity"?
max_x ||Ax||/||x||
What is the "operator norm"?
If you want to fit a polynomial to data (x_i,y_i), you can use the matrix bearing this "V" name. Just fill each row with [1,x_i,x_i^2,ldots,x_i^n]
What is a "Vandermonde matrix"?
A creative "remix" of U_r, Sigma_r, and V_r gives you this least-squares-solving matrix
What is the "(Moore-Penrose) pseudoinverse"?
Named for a titan of numerical analysis, this adaptive adjustment can help you find eigenvalues in a hurry
What is the "Wilkinson shift"?
Floating-point addition lacks this familiar "order-ignoring" property of regular addition. You might say the two don't really get along!
What is "associativity"?
We're pretty confident that QR iteration converges to this useful factorization of A
What is the "Schur decomposition"?
||x- hat x||/||x||
when f_A(x) = f(hat x)
What is "backward error"?
The SVD reveals a basis of these "fundamental subspaces" relating to a matrix A
What are the "ranges and nullspaces of A and A^**"?
Written I - 2u u^T, these transformations can help "bring home" the QR factorization
What are "Householder reflections"?
This term, which refers to the "significant digits" of a floating-point number, might also bring to mind a kind of insect
What is "the mantissa"?
This sad-sounding technique actually saves you time when you're trying to finish finding the eigenvalues of A
What is "deflation"?
max_j sum_i |a_(ij)|
What is the "matrix 1-norm"?